Radical of Lie algebra after modding out by a semisimple ideal

Asked Sep 16 '18 at 22:36

Active Sep 17 '18 at 22:09

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Let $\mathfrak{g}$ be a Lie algebra, then Levi's decomposition theorem affirms that we can decompose $\mathfrak{g} = \mathfrak{r(g)}\oplus \mathfrak{s}$, where $\mathfrak{r(g)}$ is the radical of $\mathfrak{g}$, and $\mathfrak{s}$ is a semisimple subalgebra of $\mathfrak{g}$.

I think this is easy, but I'm very stuck on this question

Question: If $\mathfrak{g}$ is a Lie algebra, with Levi's decomposition $\mathfrak{g} = \mathfrak{r(g)}\oplus \mathfrak{s}$, and $\mathfrak{s_0}$ is a semisimple ideal of $\mathfrak{g}$. Is it true that the Levi's decompostion of $\mathfrak{g}/\mathfrak{s_0} = \mathfrak{r_1}\oplus s_1$, satisfyies $\mathfrak{r(g)} \cong\mathfrak{r_1}$ ($\cong$ is the isomorfism of Lie Algebras)?

Can anyone help me?

edited Sep 17 '18 at 22:09

YCor

18,715

asked Sep 16 '18 at 22:36

Matheus Manzatto

3,271

Does this really have to do with the Levi decomposition? You're asking whether the radical of $\mathfrak{g} / \mathfrak{s}_0$ is isomorphic to the radical of $\mathfrak{g}$, right? – darij grinberg Sep 16 '18 at 22:50
And in that case, I think the answer is positive, as a particular case of https://math.stackexchange.com/questions/2196404/radical-of-a-quotient-lie-algebra . – darij grinberg Sep 16 '18 at 22:51
@darijgrinberg you are right. – Matheus Manzatto Sep 16 '18 at 22:51
@darijgrinberg do you know why in the answer linked, here can write $ X=R(L)+X∩S$? In general, isn't true that $(V_1 \oplus V_2) \cap W = V_1 \cap W \oplus V_2 \cap W.$ – Matheus Manzatto Sep 16 '18 at 23:59
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Use both $L = R\left(L\right) + S$ and $R\left(L\right) \subseteq X$. – darij grinberg Sep 17 '18 at 00:07

Radical of Lie algebra after modding out by a semisimple ideal

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